Timeline for Liouville's theorem for the submanifold of given conserved quantities?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 7, 2023 at 0:09 | comment | added | Quillo | Related/worth checking: physics.stackexchange.com/q/167556/226902 | |
| Jul 13, 2020 at 3:32 | history | edited | Qmechanic♦ |
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| Jul 12, 2020 at 22:13 | answer | added | Daniel | timeline score: 3 | |
| Jul 10, 2020 at 16:17 | comment | added | thecakeisLie | Oh and just to answer your question, the symplectic 2-form and this double integral happen to be the same in the case because you are integrating over a 2-surface so it's the same. The best way to introduce the concept of volume forms is to think about how they transform with the jacobiam matrix. | |
| Jul 10, 2020 at 16:14 | comment | added | thecakeisLie | No worries, it is an advanced subject for sure. I think you received very good answers on the other post actually. But it is a difficult subject to first encounter. And about the non-squeezing theorem, i myself don't understand (the proof of) it because as far as i know it uses a technic from algebraic geometry called witten-gromov invariant, so like i said massive overkill to your question. If i had a better answer, i would have written a post. Sorry! | |
| Jul 10, 2020 at 15:59 | comment | added | user56834 | (To be clear, I'm not sure what $dp\land dq$ means, and I've asked the question here: physics.stackexchange.com/q/564834). Is it different from the double integral $\int f dpdq$? | |
| Jul 10, 2020 at 15:20 | comment | added | thecakeisLie | So in the simplest case (or locally if you prefer) the symplectic form is canonical i.e: $\omega = dp^i \wedge dq_i$. But if you prefer, this form is what "generates" the poisson structure. I can try to give you a full fledge answer but I suspect someone will come along before (and probably do a better job at it) | |
| Jul 10, 2020 at 11:02 | comment | added | user56834 | @guillaumeTrojani, I've tried reading about symplectic geometry, but I don't yet get how it relates to hamiltonian mechanics. What is the symplectic form here? | |
| Jul 10, 2020 at 10:52 | comment | added | thecakeisLie | en.wikipedia.org/wiki/Non-squeezing_theorem maybe is a bit overkill. But the point is this result stems from symplectic geometry. | |
| Jul 10, 2020 at 10:48 | history | asked | user56834 | CC BY-SA 4.0 |